# Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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### What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by
$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.
I know that if $\...

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### An asymptotic version of the Isoperimetric inequality

Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$.
(We do not assume any thing about its boundary).
Assume that $\gamma_n$,s are smooth simple ...

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255 views

### Integer points avoiding three on a line, four on a circle

A century ago, Dudeney asked to place $16$ pawns on a chessboard with no three
on a line:
As described by David Eppstein,1 the maximum number $g_3(n)$ points that
...

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94 views

### Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:
Let three $(A)$, $(B)$, $(...

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128 views

### A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...

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### Universal point sets for 1-plane graphs

It is a notorious open problem to find a smallest set of $N$ points that
permit any $n$-vertex planar graph to be drawn in the plane without
crossings, using only those $N$ points as vertices, and ...

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143 views

### An new equilateral triangle related to the Morley triangle

Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors:
Let $ABC$ be a triangle ...

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206 views

### Are these points known? [closed]

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively.
From my construction by GeoGebra, I found two special points as ...

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123 views

### A geometric property about certain polynomials in two variables

Assume that $p(x,y)$ is a polynomial in $\mathbb{R}[x, y]$ in the form $$ p=p_{2n}+ p_{2n-1}+\ldots +p_1+p_0$$
where $p_i$ is a homogenous polynomial of degree $i$. Moreover we assume that the last ...

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### another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$
In 1984 S.D.Berman, a Soviet mathematician, ...

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79 views

### Random walks in arrangements of lines in the plane

Let $\cal{A}_n$ be a simple arrangement of $n$ lines in $\mathbb{R}^2$.
(Simple: each pair of lines meet in a distinct point, i.e.,
no three lines pass through the same point.)
Start a random walk at ...

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**1**answer

231 views

### Is there any Menelaus-type theorem for polynomials?

Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$.
In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...

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### Do two new special points in any triangle exist?

There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.
Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...

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57 views

### 8-partition of a planar convex body by 4 concurrent lines

It is known1 that any convex body $K$ in the plane can be
partitioned into $6$ equal-area pieces by $3$ concurrent lines
which meet at a point in $K$.
Call this a $6$-partition.
This result cannot be ...

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215 views

### Minimum weight triangulation of lattice points in a circle

Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...

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75 views

### Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$

We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...

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246 views

### Visual proof of convergence for Steiner's symmetrization

I want to find a visual proof of the following fact:
For any convex figure in the plane there is a sequence of Steiner's symmetrizations that makes it arbitrary close to a circular disc.
All ...

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192 views

### Tiling with incommensurate triangles

Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...

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451 views

### Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...

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434 views

### Which finite sets could be packed into a square?

This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space.
The problem starts with a two-...

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**1**answer

199 views

### Some inequalities on chain of circle packing

By my computation, I pose a conjecture as follows and I am looking for a proof:
Conjecture: Let $(O)$ be a circle with radius $R$, and $n$ be positive integer $n\ge 3$. Construct $n$ circles $(...

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### In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $

I am looking for a proof of the inequality as follows:
Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...

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131 views

### Coloring circles in plane

We assume that all the circles in the plane are each colored with one of two colors: red or blue.
My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...

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79 views

### Coloring lines in plane

We assume that all the lines in the plane are each colored with one of two colors: red or blue. Given angle $\alpha.$
My question 1. Is there possible to get two lines with the same color and angle ...

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211 views

### A generalization of the Sawayama-Thebault theorem

1. Introduction
The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...

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### $N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...

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186 views

### Yiu's equilateral triangle-triplet points

In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...

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### An ellipse through 12 points related to Golden ratio

I am looking for a proof of the problem as follows:
Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$...

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### Comparing Different Notions of Unicoherence in the Plane

Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...

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174 views

### Creativity and the mechanization of elementary geometry

In plane geometry, it is customary to say that checking proofs
is a mechanical process but that finding new theorems
is a creative activity.
Citing J. Hadamard, "logic only sanctions the conquests of ...

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### Straight-line drawing of regular polyhedra

Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...

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### What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.

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### A compact set in the plane with small sum-set and large projections

Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?
Remark. The ...

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317 views

### Bichromatic pencils

A pencil is a collection of some lines through a point, called the center of the pencil.
If the points of the plane are colored, then call a pencil bichromatic if there is a color that is present on ...

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306 views

### A question on Mobius strip and Jordan curve

If $A\subset \Bbb R^2$ then is the following statement true?
$\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq \text{Mobius strip}\quad\iff\quad A\,\,\, \text{is a Jordan curve}.$...

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380 views

### Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to:
\begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...

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386 views

### Arranging squares without overlap

What is the smallest positive real $r\in\mathbb{R}$ with the following property?
Every finite collection of squares such that the sum of their areas equals $1$ can be arranged without overlap ...

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128 views

### Trouble with plane embedding

Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.
Are there any general results which can be ...

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### Sets of points avoiding small angles

(1) $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
...

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95 views

### Cutting a square into an infinite number of triangles constrained by two rules

Can a square be cut into an infinite number of triangles so that
a) all of them are non-similar
and
b) only a finite number of them can have a common vertex?

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### Relation of Convex Polygons to Halin Graphs

Can every Halin graph be visualized as the union of a planar, strictly convex polygon with its Voronoi Diagram?
It is true, that every union of a strictly convex planar polygon with its Voronoi ...

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152 views

### Tileability and computabilty

Let $n>2$ be an integer. We consider $n$ pairs $(x_1,y_1),\dotsc,(x_n,y_n)$ in $\mathbb{N}^2$, and the polygon defined by drawing a straight line from $(x_k, y_k)$ to $(x_{k+1},y_{k+1})$ and from $(...

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### Are two triangles with equal corresponding medians, congruent?

Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...

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107 views

### Computational Geometric Aspects of Greedy Tour Expansion

Has the following problem already been investigated from the Computational Geometry point of view and what are the results regarding worst case complexity?
Given
a finite set $\mathcal{P}...

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### Can the “Bisector” be represented by a holomorphic function?

Note:
In this question, a complex number is counted as a vector initiated from the origin.
______________________________________________________________-
Is there a holomorphic function $B:\...

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### Even Isometries in neutral Geometry

Consider a Hilbert plane as in Hartshorne's 'Euclid and beyond' (axiomatic geometry), and its group of isometries f or 'rigid motion' generated by line reflections. Call f 'even' if it is the product ...

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### recursively convex plane curves

For the lack of a better term let's call a convex simple loop $u(t)$ recursively convex if for any $n \geq 0$ the $n$-th derivative $u^{(n)}(t)$ is a convex simple loop. We conjecture that any ...

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### Rolling/width functions: Characterization?

Let $K$ be a strictly convex planar body of perimeter $1$.
Roll it along the $x$-axis from $0$ to $1$, and
define $f(x)$ to be the height of the highest point of $K$
when it touches at $(x,0)$.
So $f(...

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### Patterns in local winding number sequences

This is something of a followup to an earlier question
Can any sequence of consecutive integers be realized as winding numbers?, whose answer is Yes.
Now I would like to define a local winding number ...

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571 views

### Can any sequence of consecutive integers be realized as winding numbers?

For a closed plane curve $C$, define its sequence of winding numbers to
be the sorted list of the winding numbers of each of the distinct regions
of the plane demarcated by $C$.
For example, this ...